Higher order RG flow on the Wilson line in $\mathcal{N}=4$ SYM

TL;DR

This paper calculates the two-loop beta-function for a scalar coupling in a generalized Wilson loop in $ ext{N}=4$ SYM, revealing fixed points and higher-loop contributions, with implications for anomalous dimensions and Wilson loop expectations.

Contribution

It provides the two-loop beta-function for scalar coupling in Wilson loops and explores higher-loop effects, extending previous work in $ ext{N}=4$ SYM theory.

Findings

Beta-function has fixed points at $ ext{zeta}= ext{±}1$ and 0.

Predicted two-loop anomalous dimension of scalar field on Wilson loop.

Identified higher-loop contributions from scalar ladder graphs.

Abstract

Extending earlier work, we find the two-loop term in the beta-function for the scalar coupling $\zeta$ in a generalized Wilson loop operator of the $\mathcal{N}=4$ SYM theory, working in the planar weak-coupling expansion. The beta-function for $\zeta$ has fixed points at $\zeta=\pm 1$ and $\zeta=0$, corresponding respectively to the supersymmetric Wilson-Maldacena loop and to the standard Wilson loop without scalar coupling. As a consequence of our result for the beta-function, we obtain a prediction for the two-loop term in the anomalous dimension of the scalar field inserted on the standard Wilson loop. We also find a subset of higher-loop contributions (with highest powers of $\zeta$ at each order in `t Hooft coupling) coming from the scalar ladder graphs determining the corresponding terms in the five-loop beta-function. We discuss the related structure of the circular Wilson loop expectation value commenting, in particular, on consistency with a 1d defect version of the F-theorem. We also compute (to two loops in the planar ladder approximation) the two-point correlators of scalars inserted on the Wilson line.